The subnormal embedding of complete groups
type: Item
instance of: scholarly article
title: The subnormal embedding of complete groups
zbMATH Open document ID: 0547.20018
DOI: 10.1016/0021-8693(84)90182-0
author: John C. Lennox
author: Hermann Heineken
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: Theorem A: A finite soluble complete group A cannot be subnormally embedded in the derived subgroup of a finite group G. For \(A=S_ 3\), the symmetric group of degree 3, this is a result of P. Hall (\textit{R. S. Dark} [J. Lond. Math. Soc. 43, 387-390 (1968; Zbl 0157.054)]); this proof extends to the case where A is the holomorph of a cyclic 3-group. The general result is obtained from a structure theorem (Theorem C) for a directly indecomposable, finite, complete, subnormal subgroup A of a group G where A is not the holomorph of any cyclic 3-group: set \(A^*=ter\min al\) member of the lower central series of A, \(F=Fitting\) subgroup of \(A^ G\), then (a) \(A^*\) is normal in \(A^ G\), and \((A^*)^ G\) is the direct product of all conjugates of A, (b) AF is normal in \(A^ G\), and \(A^ G/F\) is the direct product of all conjugates of AF/F in G/F, (c) \(N_{G/F}(AF/F)=(AF/F)C_{G/F}(AF/F),\) (d) if G is finite, there is a supplement U of \(A^ G\) such that \(U\cap A^ G=F\).
Mathematics Subject Classification ID: 20D35
Mathematics Subject Classification ID: 20D25
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20D45
zbMATH DE Number: 3871629
zbMATH Keywords: finite soluble complete group
zbMATH Keywords: subnormal subgroup
zbMATH Keywords: holomorph
zbMATH Keywords: lower central series
zbMATH Keywords: Fitting subgroup
MaRDI profile type: Publication
full work available at URL: 0021-8693(84)90182-0
OpenAlex ID: W2092802746
cites work: On Subnormal Embedding Theorems for Groups
cites work: Some finite solvable groups with no outer automorphisms
cites work: Endliche Gruppen I
cites work: Vertauschbare nachinvariante Untergruppen
description: scientific article; zbMATH DE number 3871629